From Blangiardo and Cameletti (2015), Section 4.7
Include derivations of \(\tilde{p}(\boldsymbol{\psi}|\mathbf{y})\), \(\tilde{p}(\theta_{i}|\boldsymbol{\psi},\mathbf{y})\), and \(\tilde{p}(\theta_{i}|\mathbf{y})\); and explanations of mode-finding and CCD.
Blangiardo and Cameletti (2015) section 4.9
The posterior distribution of the nuisance parameter is
\[p(\psi|\mathbf{y}) \propto \frac{p(\mathbf{y} | \theta, \psi) p(\theta) p(\psi)} {p(\theta | \psi, \mathbf{y})}\]
Blangiardo, Marta, and Michela Cameletti. 2015. Spatial and Spatio-Temporal Bayesian Models with R - Inla. Wiley.